Copied to
clipboard

## G = C24.42D4order 192 = 26·3

### 42nd non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24.42D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C8○D12 — C24.42D4
 Lower central C3 — C6 — C2×C12 — C24.42D4
 Upper central C1 — C2 — C2×C4 — C8.C4

Generators and relations for C24.42D4
G = < a,b,c | a8=1, b12=c2=a4, bab-1=cac-1=a3, cbc-1=b11 >

Subgroups: 352 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C4.Dic3, C2×C24, C3×M4(2), C2×Dic6, C2×D12, C4○D12, D4.3D4, C12.46D4, C12.47D4, C3×C8.C4, C8○D12, C2×C24⋊C2, C8⋊D6, C8.D6, C24.42D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q83S3, D4.3D4, C12⋊D4, C24.42D4

Character table of C24.42D4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 12 24 2 2 2 12 24 2 4 2 2 4 8 8 12 12 2 2 4 4 4 4 4 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ7 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 0 0 -1 2 2 0 0 -1 -1 -2 -2 -2 -2 2 0 0 -1 -1 -1 1 1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 0 0 2 -2 2 0 0 2 -2 2 2 -2 0 0 0 0 -2 -2 2 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 2 -2 2 0 0 2 -2 -2 -2 2 0 0 0 0 -2 -2 2 -2 -2 2 2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 0 2 2 -2 2 0 2 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 0 -1 2 2 0 0 -1 -1 2 2 2 -2 -2 0 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 2 0 2 2 -2 -2 0 2 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 0 -1 2 2 0 0 -1 -1 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 2 2 0 0 -1 2 2 0 0 -1 -1 -2 -2 -2 2 -2 0 0 -1 -1 -1 1 1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ17 2 2 -2 0 0 -1 -2 2 0 0 -1 1 2 2 -2 0 0 0 0 1 1 -1 -1 -1 1 1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ18 2 2 -2 0 0 -1 -2 2 0 0 -1 1 2 2 -2 0 0 0 0 1 1 -1 -1 -1 1 1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ19 2 2 -2 0 0 -1 -2 2 0 0 -1 1 -2 -2 2 0 0 0 0 1 1 -1 1 1 -1 -1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ20 2 2 -2 0 0 -1 -2 2 0 0 -1 1 -2 -2 2 0 0 0 0 1 1 -1 1 1 -1 -1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ21 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 0 0 0 2i -2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 0 0 0 -2i 2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 0 0 -2 4 -4 0 0 -2 2 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 4 0 0 -2 -4 -4 0 0 -2 -2 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ25 4 -4 0 0 0 4 0 0 0 0 -4 0 2√-2 -2√-2 0 0 0 0 0 0 0 0 -2√-2 2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 0 0 0 4 0 0 0 0 -4 0 -2√-2 2√-2 0 0 0 0 0 0 0 0 2√-2 -2√-2 0 0 0 0 0 0 complex lifted from D4.3D4 ρ27 4 -4 0 0 0 -2 0 0 0 0 2 0 -2√-2 2√-2 0 0 0 0 0 2√3 -2√3 0 -√-2 √-2 √-6 -√-6 0 0 0 0 complex faithful ρ28 4 -4 0 0 0 -2 0 0 0 0 2 0 2√-2 -2√-2 0 0 0 0 0 -2√3 2√3 0 √-2 -√-2 √-6 -√-6 0 0 0 0 complex faithful ρ29 4 -4 0 0 0 -2 0 0 0 0 2 0 2√-2 -2√-2 0 0 0 0 0 2√3 -2√3 0 √-2 -√-2 -√-6 √-6 0 0 0 0 complex faithful ρ30 4 -4 0 0 0 -2 0 0 0 0 2 0 -2√-2 2√-2 0 0 0 0 0 -2√3 2√3 0 -√-2 √-2 -√-6 √-6 0 0 0 0 complex faithful

Smallest permutation representation of C24.42D4
On 48 points
Generators in S48
```(1 43 7 25 13 31 19 37)(2 26 20 44 14 38 8 32)(3 45 9 27 15 33 21 39)(4 28 22 46 16 40 10 34)(5 47 11 29 17 35 23 41)(6 30 24 48 18 42 12 36)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 33 13 45)(2 44 14 32)(3 31 15 43)(4 42 16 30)(5 29 17 41)(6 40 18 28)(7 27 19 39)(8 38 20 26)(9 25 21 37)(10 36 22 48)(11 47 23 35)(12 34 24 46)```

`G:=sub<Sym(48)| (1,43,7,25,13,31,19,37)(2,26,20,44,14,38,8,32)(3,45,9,27,15,33,21,39)(4,28,22,46,16,40,10,34)(5,47,11,29,17,35,23,41)(6,30,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,33,13,45)(2,44,14,32)(3,31,15,43)(4,42,16,30)(5,29,17,41)(6,40,18,28)(7,27,19,39)(8,38,20,26)(9,25,21,37)(10,36,22,48)(11,47,23,35)(12,34,24,46)>;`

`G:=Group( (1,43,7,25,13,31,19,37)(2,26,20,44,14,38,8,32)(3,45,9,27,15,33,21,39)(4,28,22,46,16,40,10,34)(5,47,11,29,17,35,23,41)(6,30,24,48,18,42,12,36), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,33,13,45)(2,44,14,32)(3,31,15,43)(4,42,16,30)(5,29,17,41)(6,40,18,28)(7,27,19,39)(8,38,20,26)(9,25,21,37)(10,36,22,48)(11,47,23,35)(12,34,24,46) );`

`G=PermutationGroup([[(1,43,7,25,13,31,19,37),(2,26,20,44,14,38,8,32),(3,45,9,27,15,33,21,39),(4,28,22,46,16,40,10,34),(5,47,11,29,17,35,23,41),(6,30,24,48,18,42,12,36)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,33,13,45),(2,44,14,32),(3,31,15,43),(4,42,16,30),(5,29,17,41),(6,40,18,28),(7,27,19,39),(8,38,20,26),(9,25,21,37),(10,36,22,48),(11,47,23,35),(12,34,24,46)]])`

Matrix representation of C24.42D4 in GL4(𝔽73) generated by

 48 62 53 0 11 37 53 53 0 0 48 11 0 0 62 37
,
 14 66 11 10 7 7 1 11 0 14 66 7 59 0 66 59
,
 25 11 67 61 36 48 61 67 0 0 25 62 0 0 37 48
`G:=sub<GL(4,GF(73))| [48,11,0,0,62,37,0,0,53,53,48,62,0,53,11,37],[14,7,0,59,66,7,14,0,11,1,66,66,10,11,7,59],[25,36,0,0,11,48,0,0,67,61,25,37,61,67,62,48] >;`

C24.42D4 in GAP, Magma, Sage, TeX

`C_{24}._{42}D_4`
`% in TeX`

`G:=Group("C24.42D4");`
`// GroupNames label`

`G:=SmallGroup(192,457);`
`// by ID`

`G=gap.SmallGroup(192,457);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,58,1123,136,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^12=c^2=a^4,b*a*b^-1=c*a*c^-1=a^3,c*b*c^-1=b^11>;`
`// generators/relations`

Export

׿
×
𝔽